On a Relation between Euler characteristics of \MakeLowercase{de} Rham cohomology and Koszul cohomology of graded local cohomology modules

Let $K$ be a field of characteristic zero. Let $R=K[X_0,X_1,\dots ,X_n]$ be standard graded. Let $A_{n+1}(K)$ be the $(n+1{)}^{th}$ Weyl algebra over $K$. Let $I$ be a homogeneous ideal of $R$ and let $M=H_I^i(R)$ for some $i\ge 0$. By a result of Lyubeznik, $M$ is a graded holonomic $A_{n+1}(K)$-module for each $i\ge 0$. Let ${\chi }^c(\mathbf{\partial },M)$ (${\chi }^c(\mathbf{X},M)$) be the Euler characteristics of de Rham cohomology (resp. Koszul cohomology) of $M$. We prove ${\chi }^c(\mathbf{\partial },M)=(-1{)}^{n+1}{\chi }^c(\mathbf{X},M)$.